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We construct certain exotic embeddings of the  n -torus      T n {T^n}       in      T n + 2 {T^{n + 2}}       in the standard homotopy class. We turn an embedding      f : T n →  T n + 2 f:{T^n} \to {T^{n + 2}}       characteristic if there exists some map      α  : T n + 2 →  T n + 2 \alpha :{T^{n + 2}} \to {T^{n + 2}}       in the standard homotopy class with the property that      α  ∘  f : T n →  T n + 2 \alpha \; \circ \;f:{T^n} \to {T^{n + 2}}       is the standard coordinate inclusion and      α  ( T n + 2 −  f ( T n ) ) ⊂  T n + 2 −  T n \alpha ({T^{n + 2}} - f({T^n})) \subset {T^{n + 2}} - {T^n}      . We find examples of noncharacteristic embeddings,  f , in dimensions      n = 4 k + 1 n = 4k + 1      ,      n ≥  5 n \geq 5      , and show that these examples are not even cobordant to characteristic embeddings. We let  G  denote the fundamental group of the complement of the standard coordinate inclusion,      T n + 2 −  T n {T^{n + 2}} - {T^n}      . Then we can associate to  f  a real-valued signature function on the set of  j -dimensional unitary representations of      G ¯  \bar G      , where      G ¯  \bar G       denotes the fundamental group of the localization of      T n + 2 −  T n {T^{n + 2}} - {T^n}       with respect to homology with local coefficients in      Z [ Z n + 2 ] \mathbb {Z}[{\mathbb {Z}^{n + 2}}]      . This function is a cobordism invariant which has certain periodicity properties for characteristic embeddings. We verify that this periodicity does not hold for our examples,  f , implying that they are not characteristic. Additional results include a proof that the examples,  f , become cobordant to characteristic embeddings upon taking the cartesian product with the identity map on a circle.

Noncharacteristic embeddings of the 𝑛-dimensional torus in the (𝑛+2)-dimensional torus